
where \(:_+\rightarrow _+\)  is the same renormalization as Section REF .

The similarity with OT stems from the use of the constraint set \((,)\) .
The key difference is the quadratic dependence in the plan \(\pi \otimes \pi \) .
This quadratic dependence is what characterizes it as a QAP, and makes it an NP-hard problem to solve in general [1]}.
It can be interpreted similarly to an OT program which maps pairs \((x,x^{\prime })\)  to \((y,y^{\prime })\)  such that \(d_(x,x^{\prime })\)  is close to \(d_(y,y^{\prime })\) , under the constraint that \(\pi \)  is a transport plan between \((,)\) .
The choice of \(\)  is crucial for the metric properties of GW.
It is proved in [2]} that \(^{1/q}\)  is a distance when \((t)=t^q\)  (\(q\ge 1\) ).
